A great challenge within ALE FEM is to ensure energy stability on the discrete level. If a scheme is stable in the energy norm, then there is no spurious energy brought into the system due to discretization. Such schemes are relatively easy to construct for the problems on fixed meshes, but significantly more challenging to derive on moving meshes. Energy stable schemes are often studied in context of fluid–structure interaction (FSI) problems modeled within ALE framework (see, e.g., [1]}, [2]}, [3]} and references therein). For FSI problems, energy balance of ALE FEM scheme is usually based on two main ingredients (within monolithic approach): space conservation law (SCL) and proper coupling of fluid and structure ([3]}). In case discrete SCL is not satisfied, artificial energy sinks and/or sources appear in the numerical scheme, and spurious energy is brought into the system. More generally, SCL is closely intertwined with the stability of moving mesh methods – it roots from the Reynolds transport theorem when transiting from the continuous to the discrete level. More details on SCL and stability issues may be found in [5]}, [6]} and references therein. In context of simulating the free surface flows within ALE framework, SCL plays the same role for numerical stability as in the case of FSI. For the case of the free surface flows, however, sources of spurious energies are also known to appear on fluid–fluid interface, fluid–solid interface, and triple contact line.
